Reflections Reflections

What are reflections?What are reflections?

What are the characteristics of What are the characteristics of a reflection?a reflection?

A reflection is a A reflection is a mirror imagemirror image of the original shape.of the original shape.

This means it looks the same, This means it looks the same, except that it is except that it is flippedflipped!!

A reflected figure is A reflected figure is congruentcongruent to its original to its original shape.shape.

How do we reflect a figure over How do we reflect a figure over the y-axis?the y-axis?

4 units away

So, A ’ shouldBe 4 units awayFrom the y-axis! 4 units away

Plot ΔABC:A(-4, 4)B(-2, 4)C(-4, 1)

What are the CoordinatesOf ΔA ’B ’C ’?

A ’(4, 4)

B ’(2, 4)

C ’(4, 1)

Observations:

1. Both triangles are congruent

2. The reflected figure is the mirror image of the original.

3. The reflected figure flippedflipped over to the right.

4. The x coordinate switch to the opposite value.

2 units2 units 2 2 unitsunits

4 units away 4 units away

A

B

C

A ’B ’

Reflection over the y – axis.Reflection over the y – axis.

Graph the ΔABC:

A(-4, 2)B(-3, -1)C(-5, -2)

A

B ’

A ’

C ’

BC

4 units 4 units

3 units 3 units

5 units 5 units

CoordinatesOf ΔA ’B ’C ’:

A ’ (4, 2)

B ’(3, -1)

C ’(5, -2)

Reflection Over the x-axisReflection Over the x-axisGraph the trapezoid:

A(-4, 4)

B(-1, 5)

C(-1, 1)

D(-4, 2)

C

AB

D

B ’

D ’

A ’

C’

A ’(-4, -4)

B ’(-1, -5)

C ’(-1, -1)

D ’(-4, -2)

Observations:

•Figures are CONGRUENT.

•Y values change to the opposite.

•The distance from the line of reflection stays the same for each shape: Example:Example:C is 1 unit from the x-axisC ’ is1 unit from the x-axis

Reflecting over the X-axisReflecting over the X-axis

Graph the ΔABC:

A(3, 6)

B(-6, -1)

C(5, 1)

A ’(3, -6)

B ’(-6, -1)

C ’(5, -1)

A

B

C

A ’

B ’

C ’

What do we know about What do we know about reflections so far?reflections so far?

The figures are CONGRUENT. This means The figures are CONGRUENT. This means they are the same size and shape.they are the same size and shape.

Distance from the line of reflection stays Distance from the line of reflection stays the same. For example, if point A is 2 the same. For example, if point A is 2 units from the reflection line, then Aunits from the reflection line, then A’’ is is also 2 units from that line. It is only going also 2 units from that line. It is only going in the opposite direction.in the opposite direction.

The reflected figure is the The reflected figure is the mirror imagemirror image of the original shape. It is only of the original shape. It is only flippedflipped..

Reflecting over the line Reflecting over the line y=xy=x

First, what is the line y=x and how do we find it?!First, what is the line y=x and how do we find it?!1.1. In order to graph a line, we need coordinate In order to graph a line, we need coordinate

points. Which means we need a t-chart.points. Which means we need a t-chart.Pick numbers to go in for your x values.Pick numbers to go in for your x values.

2. Then solve for the y values by substituting x 2. Then solve for the y values by substituting x into your equation y =x.into your equation y =x.

XX

YY

-3-3 -2-2 -1-1 00 11 22 33

-3-3 -2-2 -1-1 00 11 22 33

Draw the line y =x in the graph Draw the line y =x in the graph by plotting the points and by plotting the points and

highlight it!highlight it!

Graph the ΔABC:

A(-4, 2)

B(-3, -1)

C(-5, -2)

A

B

C

Using the mirror, place it on the Line y =x. Then look throughThe mirror to reflect the points.

A ’

B ’

C ’

A ’(2, -4)

B ’(-1, -3)

C ’(-2, -5)

What do you noticeWhat do you noticeAbout the coordinate About the coordinate Points?Points?

Reflect the following over the Reflect the following over the line y =xline y =x

Graph the heart given the Following coordinates:

A(4, - 6)

B(7, -3)

C(6, -1)

D(5, -1)

E(4, -2)

F(3, -1)

G(2, -1)

H(1, -3)

A ’(-6, 4)

B ’(-3, 7)

C ’(-1, 6)

D ’(-1, 5)

E ’(-2, 4)

F ’(-1, 3)

G ’(-1, 2)

H ’(-3, 1)

Practice ProblemsPractice Problems

For the following For the following FIVE problems, FIVE problems, graph the graph the reflection and reflection and state the state the coordinates.coordinates.

Practice ProblemsPractice Problems

Solution to Problem 1Solution to Problem 1

Problem TwoProblem Two

Solution to Problem 2Solution to Problem 2

Problem ThreeProblem Three

Solution to Problem ThreeSolution to Problem Three

Practice Problem 4Practice Problem 4

Solution to Problem 4Solution to Problem 4

Practice Problem 5Practice Problem 5

Solution to Problem 5Solution to Problem 5

HomeworkHomework

Homework SolutionHomework Solution